The computations performed in the routine
noetherNormalization use a random linear change of coordinates, hence one should expect the output to change each time the routine is executed.
i1 : R = QQ[x_1..x_4];
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i2 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o2 : Ideal of R
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i3 : (f,J,X) = noetherNormalization I
3 2 11 2
o3 = (map(R,R,{-x + x + x , x , 4x + -x + x , x }), ideal (--x + x x +
8 1 2 4 1 1 3 2 3 2 8 1 1 2
------------------------------------------------------------------------
3 3 17 2 2 2 3 3 2 2 2 2 2
x x + 1, -x x + --x x + -x x + -x x x + x x x + 4x x x + -x x x
1 4 2 1 2 4 1 2 3 1 2 8 1 2 3 1 2 3 1 2 4 3 1 2 4
------------------------------------------------------------------------
+ x x x x + 1), {x , x })
1 2 3 4 4 3
o3 : Sequence
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The next example shows how when we use the lexicographical ordering, we can see the integrality of
R/ f I over the polynomial ring in
dim(R/I) variables:
i4 : R = QQ[x_1..x_5, MonomialOrder => Lex];
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i5 : I = ideal(x_2*x_1-x_5^3, x_5*x_1^3);
o5 : Ideal of R
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i6 : (f,J,X) = noetherNormalization I
2 10 4 1
o6 = (map(R,R,{-x + 2x + x , x , --x + x + x , -x + -x + x , x }),
5 1 2 5 1 9 1 2 4 5 1 4 2 3 2
------------------------------------------------------------------------
2 2 3 8 3 24 2 2 12 2 24 3
ideal (-x + 2x x + x x - x , ---x x + --x x + --x x x + --x x +
5 1 1 2 1 5 2 125 1 2 25 1 2 25 1 2 5 5 1 2
------------------------------------------------------------------------
24 2 6 2 4 3 2 2 3
--x x x + -x x x + 8x + 12x x + 6x x + x x ), {x , x , x })
5 1 2 5 5 1 2 5 2 2 5 2 5 2 5 5 4 3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 10x_1x_2x_5^6-96x_2^9x_5-320x_2^9+24x_2^8x_5^2+160x_2^8x_5-4x_2^
{-9} | 400x_1x_2^2x_5^3-30x_1x_2x_5^5+200x_1x_2x_5^4+288x_2^9-72x_2^8x_
{-9} | 160000x_1x_2^3+12000x_1x_2^2x_5^2+160000x_1x_2^2x_5+90x_1x_2x_5^
{-3} | 2x_1^2+10x_1x_2+5x_1x_5-5x_2^3
------------------------------------------------------------------------
7x_5^3-80x_2^7x_5^2+40x_2^6x_5^3-20x_2^5x_5^4+10x_2^4x_5^5+50x_2^2x_5^6
5-160x_2^8+12x_2^7x_5^2+160x_2^7x_5-120x_2^6x_5^2+60x_2^5x_5^3-30x_2^4x
5-300x_1x_2x_5^4+4000x_1x_2x_5^3+40000x_1x_2x_5^2-864x_2^9+216x_2^8x_5+
------------------------------------------------------------------------
+25x_2x_5^7
_5^4+200x_2^4x_5^3+2000x_2^3x_5^3-150x_2^2x_5^5+2000x_2^2x_5^4-75x_2x_
720x_2^8-36x_2^7x_5^2-600x_2^7x_5+800x_2^7+360x_2^6x_5^2-1200x_2^6x_5-
------------------------------------------------------------------------
5^6+500x_2x_5^5
8000x_2^6-180x_2^5x_5^3+600x_2^5x_5^2+4000x_2^5x_5+80000x_2^5+90x_2^4x_5
------------------------------------------------------------------------
^4-300x_2^4x_5^3+4000x_2^4x_5^2+40000x_2^4x_5+800000x_2^4+60000x_2^3x_5^
------------------------------------------------------------------------
2+1200000x_2^3x_5+450x_2^2x_5^5-1500x_2^2x_5^4+50000x_2^2x_5^3+600000x_2
------------------------------------------------------------------------
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^2x_5^2+225x_2x_5^6-750x_2x_5^5+10000x_2x_5^4+100000x_2x_5^3 |
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5 1
o7 : Matrix R <--- R
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If
noetherNormalization is unable to place the ideal into the desired position after a few tries, the following warning is given:
i8 : R = ZZ/2[a,b];
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i9 : I = ideal(a^2*b+a*b^2+1);
o9 : Ideal of R
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i10 : (f,J,X) = noetherNormalization I
--warning: no good linear transformation found by noetherNormalization
2 2
o10 = (map(R,R,{a + b, a}), ideal(a b + a*b + 1), {b})
o10 : Sequence
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Here is an example with the option
Verbose => true:
i11 : R = QQ[x_1..x_4];
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i12 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o12 : Ideal of R
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i13 : (f,J,X) = noetherNormalization(I,Verbose => true)
--trying random transformation: 1
--trying with basis element limit: 5
--trying with basis element limit: 20
2 9 9 11 2
o13 = (map(R,R,{-x + 2x + x , x , --x + -x + x , x }), ideal (--x +
9 1 2 4 1 10 1 8 2 3 2 9 1
-----------------------------------------------------------------------
1 3 41 2 2 9 3 2 2 2 9 2
2x x + x x + 1, -x x + --x x + -x x + -x x x + 2x x x + --x x x
1 2 1 4 5 1 2 20 1 2 4 1 2 9 1 2 3 1 2 3 10 1 2 4
-----------------------------------------------------------------------
9 2
+ -x x x + x x x x + 1), {x , x })
8 1 2 4 1 2 3 4 4 3
o13 : Sequence
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The first number in the output above gives the number of linear transformations performed by the routine while attempting to place
I into the desired position. The second number tells which
BasisElementLimit was used when computing the (partial) Groebner basis. By default,
noetherNormalization tries to use a partial Groebner basis. It does this by sequentially computing a Groebner basis with the option
BasisElementLimit set to predetermined values. The default values come from the following list:
{5,20,40,60,80,infinity}. To set the values manually, use the option
LimitList:
i14 : R = QQ[x_1..x_4];
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i15 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o15 : Ideal of R
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i16 : (f,J,X) = noetherNormalization(I,Verbose => true,LimitList => {5,10})
--trying random transformation: 1
--trying with basis element limit: 5
--trying with basis element limit: 10
8 6 11 2 6
o16 = (map(R,R,{-x + -x + x , x , x + 4x + x , x }), ideal (--x + -x x
3 1 7 2 4 1 1 2 3 2 3 1 7 1 2
-----------------------------------------------------------------------
8 3 242 2 2 24 3 8 2 6 2 2
+ x x + 1, -x x + ---x x + --x x + -x x x + -x x x + x x x +
1 4 3 1 2 21 1 2 7 1 2 3 1 2 3 7 1 2 3 1 2 4
-----------------------------------------------------------------------
2
4x x x + x x x x + 1), {x , x })
1 2 4 1 2 3 4 4 3
o16 : Sequence
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To limit the randomness of the coefficients, use the option
RandomRange. Here is an example where the coefficients of the linear transformation are random integers from
-2 to
2:
i17 : R = QQ[x_1..x_4];
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i18 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o18 : Ideal of R
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i19 : (f,J,X) = noetherNormalization(I,Verbose => true,RandomRange => 2)
--trying random transformation: 1
--trying with basis element limit: 5
--trying with basis element limit: 20
2
o19 = (map(R,R,{- 2x - x + x , x , 2x - 2x + x , x }), ideal (- x - x x
1 2 4 1 1 2 3 2 1 1 2
-----------------------------------------------------------------------
3 2 2 3 2 2 2
+ x x + 1, - 4x x + 2x x + 2x x - 2x x x - x x x + 2x x x -
1 4 1 2 1 2 1 2 1 2 3 1 2 3 1 2 4
-----------------------------------------------------------------------
2
2x x x + x x x x + 1), {x , x })
1 2 4 1 2 3 4 4 3
o19 : Sequence
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This symbol is provided by the package NoetherNormalization.